One of the most intriguing mysteries in science is the origin of life. Despite extensive research in this area, we are still far from understanding how life has emerged on Earth. One promising hypothesis is the RNA world theory (Gilbert, 1986; Doudna and Szostak, 1989; Orgel, 2004; Lincoln and Joyce, 2009; Robertson and Joyce, 2012; Higgs and Lehman, 2015), inspired by the discovery of ribozymes (Kruger et al., 1982), that is, RNA molecules capable of enzymatic activity. According to this hypothesis, all processes in early life were carried out by the RNA, which was used both to store information and catalyze biochemical reactions. In particular, specific ribozymes could have catalyzed the self-replication of arbitrary RNA sequences, a function currently performed by specialized protein-based enzymes. However, based on the results of the existing experiments (Bartel and Szostak, 1993; Horning and Joyce, 2016), such a catalytic function requires rather long and carefully designed RNA sequences, which are highly unlikely to arise spontaneously. In contrast, one of the simplest catalytic activities of ribozymes is their ability to cleave an RNA sequence at a specific site. Indeed, such ribozymes independently evolved in multiple branches of life (de la Peña and García-Robles, 2010) and have been shown to emerge rapidly and repeatedly from artificial selection (Williams et al., 1995; Salehi-Ashtiani and Szostak, 2001). DNA molecules have also been shown to be capable of site-specific cleavage targeting either RNA (Breaker and Joyce, 1994) or DNA sequences (Silverman, 2005; Chandra et al., 2009).

In this article, we consider a population of information-carrying polymers capable of templated non-enzymatic replication (Szostak, 2012; Kim et al., 2021). This may have been the state of the proto-RNA world before the emergence of ribozymes. This could involve heteropolymers chemically distinct from present-day RNA (Kim et al., 2021) and/or inorganic catalysts such as mineral surfaces (Ferris, 2005; Jerome et al., 2022). We demonstrate that (i) even spontaneous cleavage promotes replication by generating short fragments used as primers for templated growth and (ii) catalytic cleavage activity naturally emerges in these populations and gets selected by the evolution (Lukin, 2010).

In a series of previous studies, we have shown that non-enzymatic templated replication can lead to the formation of longer chains (Tkachenko and Maslov, 2015) as well as to a reduction in sequence entropy (Tkachenko and Maslov, 2018). Such a reduction in entropy has subsequently been observed experimentally for templated ligation of DNA oligomers (Kudella et al., 2021). This selection in sequence space can be seen as a first step toward Darwinian evolution. However, this does not necessarily imply the emergence of a catalytic function. In this article, we build on these findings and further investigate the potential for the evolution of catalytic activity in the proto-RNA world.

In our model, we consider the population dynamics of a pool of heteropolymers analogous to the familiar nucleic acids (RNA or DNA) but capable of enzyme-free templated polymerization. The basic processes in this scenario are similar to those in polymerase chain reaction (PCR), where the system is driven out of equilibrium by cyclic changes in the environment (e.g., temperature), which we refer to as ‘night’ and ‘day’ phases. During the night phase, heteropolymers hybridize with each other following Watson–Crick-like complementarity rules. If the terminus of one chain hybridizes with the middle section of another chain, the former can be gradually elongated by the virtue of non-enzymatic templated polymerization (Sulston et al., 1968; Weimann et al., 1968; Lohrmann et al., 1980; Duzdevich et al., 2020). We will refer to this type of hybridization as ‘end-to-middle’, the former chain as the ‘primer’ and the latter chain as the ‘template’. During the day phase, the hybridized structures melt and all the heteropolymers separate from each other. During the next night, they hybridize with new partners, providing them with the opportunity to elongate further. Unlike the classical PCR process, we assume that the polymerization in our system occurs without any assistance from enzymes and may proceed in either direction along the chain. Equivalently, instead of polymerization, the elongation could rely on ligation with very short chain segments. It is important to note that in the context of RNA, such bidirectional elongation requires chemical activation of the phosphate group at the 5′ end of the primer to provide free energy for the newly formed covalent bond. Like the polymerization process itself, achieving this without enzymes is biochemically challenging. One might speculate that prebiotic evolution relied on inorganic catalysis, such as on mineral surfaces, or involved polymers other than today’s RNA.

The elongation of primers naturally leads to the copying of information from the template’s sequence. The obvious limiting factor for this process is the availability of primers and the likelihood of end-to-middle hybridization resulting in elongation. The key observation behind our model is that the breakup (cleavage) of a chain creates a new pair of potential primers. Each of them could be elongated during subsequent nights. Thus, somewhat counterintuitively, breakup of chains results in their proliferation.

Our previous theoretical (Tkachenko and Maslov, 2015; Tkachenko and Maslov, 2018) and experimental (Kudella et al., 2021) results demonstrated that templated-assisted replication of heteropolymers has a generic tendency to substantially reduce their sequence entropy. Such reduction has important consequences in the context of the current work: it significantly increases the likelihood of end-to-middle hybridization of chains during the night phase. That in turn creates an evolutionary pressure to further decrease sequence entropy. A detailed study of this fascinating mechanism falls beyond the scope of the current study. However, below we will assume the logical end of this dynamics where the pool of sequences is composed of fragments of one or several nearly non-overlapping master sequences and their complementaries. Note that for any two overlapping chains, such that the sequence of the first one is a fragment of the master sequence while the other is a fragment of the complementary master sequence, the end-to-middle binding is essentially guaranteed. The exception to this rule is when both chains terminate at the same points so that they are exact complements of each other.

Random cleavage model

Based on the argument presented above, we focus on the case of a system populated with chains that are fragments of a single master sequence or its complement. We denote the total concentration of fragments of the master sequence as ${\displaystyle c(t)}$, while the concentration of all fragments of the complementary sequence as ${\displaystyle \overline{c}(t)}$. Our system operates in a chemostat, that is, a reservoir constantly supplied with fresh monomers at the concentration ${\displaystyle m_0}$ and diluted at the rate ${\displaystyle \delta }$. Let ${\displaystyle M(t)}$ (respectively ${\displaystyle \overline{M}(t)}$) be the concentration of all monomers incorporated into chains of the subpopulation ${\displaystyle c(t)}$ (respectively ${\displaystyle \overline{c}(t)}$). The concentration of free monomers not incorporated into any chains is given by:

(1) ${\displaystyle m=m_0-M-\overline{M}}$

It is convenient to introduce a minimal length ${\displaystyle l_0}$ of a chain that would hybridize with its complementary partner during the night phase, and use it as the unit of chain length, instead of a single monomer. In effect, this leads to the renormalization of all monomer concentrations as ${\displaystyle m=[m]/l_0}$, where ${\displaystyle [m]}$ is the conventional molarity. ${\displaystyle M}$, ${\displaystyle \overline{M}}$, and ${\displaystyle m_0}$ are similarly renormalized, while the polymer concentration remains unmodified: ${\displaystyle c=[c]}$. In what follows, we renormalize all lengths and concentrations so that ${\displaystyle l_0=1}$.

We assume that nights are sufficiently long and that the hybridization rate is sufficiently fast so that early on during each night phase most chains find partners with a complementary overlap. That assumes that the total concentration of all master sequence fragments, ${\displaystyle c}$, is lower than the total concentration of all fragments in the complementary subpopulation, ${\displaystyle \overline{c}}$. If this is not the case, that is, if ${\displaystyle c>\overline{c}}$, only a fraction ${\displaystyle \overline{c}/c}$ of all fragments of the master sequence find a partner, while the rest of the ${\displaystyle c}$-subpopulation remains unpaired and thus does not elongate. For simplicity, our model neglects the possibility of the formation of hybridized complexes involving more than two chains.

It is well known that self-replication based on template-based polymerization or ligation is vulnerable to product inhibition, that is, re-hybridization of the products intended to act as templates (Szostak, 2012; Tupper and Higgs, 2021). Indeed, since full-length templates and their complements would bind more strongly to each other than any shorter fragment, the primers would typically be displaced from the templates by longer chains, leading to effective ‘template’ poisoning. In our model, however, the master sequence fragments act as both primers and templates. Of two oligomers in a hybridized pair, the one that extends further in a particular direction acts as a template for the growth of the other. At the same time, the partner may also act as a template for the growth of the first oligomer in the other direction. In either case, there will typically be two growing ends per duplex. Full inhibition occurs only when two hybridized chains terminate at exactly the same point in both directions. Since this is a rather rare event, we neglect its effect.

As discussed earlier, two hybridized chains whose sequences are fragments of the master sequence and its complementary would typically have two ends that undergo templated growth at a certain rate proportional to monomer concentration ${\displaystyle m}$ (see Figure 1). The exception to this rule is when these two hybridized chains terminate at exactly the same point. In our model, the average primer elongation rate is ${\displaystyle r\cdot m(t)}$. The rate parameter ${\displaystyle r}$ accounts for the finite probability of primer binding to a template during the night phase as well as for a finite night-to-day ratio. Note that the value of ${\displaystyle r}$ does not change after our renormalization ${\displaystyle l_0\to 1}$. The ${\displaystyle M(t)}$ dynamics is thus given by

(2) $\frac{dM}{dt}=rm\cdot \mathrm{m}\mathrm{i}\mathrm{n}(c,\overline{c})-\delta \cdot M$

Figure 1

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The equation for the complementary subpopulation is obtained by replacing ${\displaystyle M}$ with ${\displaystyle \overline{M}}$. Note that ${\displaystyle m}$ includes only monomers, and therefore is not subdivided into two complementary subpopulations.

We assume that the breakup of chains is completely random and happens at a constant rate ${\displaystyle {\beta }_0}$ at any internal bond along the chain. The concentration of these breakable bonds is given by ${\displaystyle M}$. Because of our choice of the unit length ${\displaystyle l_0}$, ${\displaystyle {\beta }_0=l_0{\beta }_0^{\mathrm{\prime }}}$, where ${\displaystyle {\beta }_0^{\mathrm{\prime }}}$ is the cleavage rate of a single bond. Since each fragmentation of a chain creates one new primer, the equation governing the overall concentration of chain fragments in the subpopulation ${\displaystyle c}$ is given by

(3) ${\displaystyle \frac{dc}{dt}={\beta }_{0}M-\delta c}$

Once again, the equation for the complementary subpopulation is obtained by replacing ${\displaystyle c}$ with ${\displaystyle \overline{c}}$ and ${\displaystyle M}$ with ${\displaystyle \overline{M}}$.

Combining Equations 2 and 3, we observe that the steady state is a symmetric mixture ${\displaystyle c=\overline{c}}$. The average length (in units of ${\displaystyle l_0}$) of all chain fragments in the subpopulation ${\displaystyle c}$ is given by

(4) ${\displaystyle ⟨L⟩=\frac{M}{c}=\frac{\delta }{{\beta }_0}=\frac{r{m}^{\ast }}{\delta }.}$

This in turn determines the steady-state concentrations:

(5) ${\displaystyle m^{\ast }=\frac{{\delta }^2}{{\beta }_{0}r}}$

(6) ${\displaystyle c=\overline{c}=\frac{1}{2}\left(\frac{{\beta }_0}{\delta }{m}_0-\frac{\delta }{r}\right)}$

To obtain Equation 6, we combined Equations 1 and 4. Note that the mutually templating chains survive only when the concentration of free monomers supplied to the system ${\displaystyle m_0}$ exceeds ${\displaystyle m^{\ast }}$.

System dynamics

A crucial feature of this multicomponent system is that catalytic cleavage depends on the cooperativity between all four subpopulations ${\displaystyle a}$, ${\displaystyle \overline{a}}$, ${\displaystyle b}$, and ${\displaystyle \overline{b}}$. To understand when such a cooperative steady state exists, we have numerically solved Equations 7–11 for ${\displaystyle {\beta }_0=0.015}$, ${\displaystyle \lambda =2}$, ${\displaystyle \delta =1}$ and different values of the catalytic cleavage rate ${\displaystyle \beta }$. Figure 2A–C show the dynamic trajectories of our system in a–b space for a wide range of initial conditions. For ${\displaystyle \beta }$ too small (e.g., ${\displaystyle \beta =6}$ in Figure 2A) or too large (e.g., ${\displaystyle \beta =18}$ in Figure 2C), the only steady-state solutions correspond to the survival of either the ${\displaystyle a/\overline{a}}$ or ${\displaystyle b/\overline{b}}$ subpopulation. In these non-cooperative fixed points, marked with red and blue stars in Figure 2, one set of chains drives the other to extinction. Thus, they propagate by random rather than catalytic cleavage. The concentrations of monomers and a complementary pair of surviving chains at a non-cooperative fixed point are given by Equations 5 and 6, respectively.

Figure 2

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For an intermediate value of β (e.g., for ${\displaystyle \beta =10}$ shown in Figure 2B), we observe the emergence of a new cooperative fixed point (marked by the green star), where all four subpopulations survive at concentrations ${\displaystyle a=\overline{a}>0}$ and ${\displaystyle b=\overline{b}>0}$. This fixed point is mainly maintained by catalytic cleavage.

The phase portrait of our system shown in Figure 2B suggests a plausible scenario for the emergence and subsequent evolution of this cooperative system. This evolutionary scenario starts with ${\displaystyle b/\overline{b}}$ subpopulations existing alone. Eventually, copying errors might result in the emergence of a small subpopulation of ${\displaystyle a}$ or ${\displaystyle \overline{a}}$ at a concentration ${\displaystyle a\ll b}$. If this concentration is greater than a certain threshold separating blue and green trajectories in Figure 2B, the dynamics of the system would drive it to the cooperative fixed point (green star in Figure 2B). For specific parameters used to generate Figure 2B, the minimal ratio between concentrations ${\displaystyle a}$ and ${\displaystyle b}$ is roughly 0.01.

Properties of the cooperative steady state

To better understand the conditions for the existence of the cooperative regime, we analytically derived the steady-state solutions of Equations 7–11. The key result is the relationship between the steady-state monomer concentration, ${\displaystyle m_{\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{p}}^{\ast }}$, and the catalytic cleavage rate β for given values of ${\displaystyle {\beta }_0}$, λ, δ, and ${\displaystyle r}$ derived in Appendix 1. Figure 3A shows this dependence for ${\displaystyle \lambda =2}$, and different values of ${\displaystyle {\beta }_0/\delta }$ alongside with data points (open circles) obtained by direct numerical solution of dynamical (Equations 7–11).

Figure 3

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The stable fixed point corresponds to the monotonically increasing branch of the graph ${\displaystyle m_{\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{p}}^{\ast }}$ vs. ${\displaystyle \beta }$ (solid lines in Figure 3A), while two decreasing branches (dashed lines in Figure 3A) correspond to two dynamically unstable saddle points separating different steady-state solutions (see Figure 2B).

Note that the stability of our cooperative fixed point is a non-trivial result. For example, in a related model by Kamimura et al., 2019, the fixed point corresponding to a viable composite replicase is dynamically unstable and requires additional stabilization, for example, by cell-like compartments.

Increasing the parameter ${\displaystyle {\beta }_0/\delta }$, for example, by making the dilution rate ${\displaystyle \delta }$ smaller, makes the range of ${\displaystyle \beta }$ for which the cooperative solution exists progressively smaller until it altogether disappears above ${\displaystyle {\beta }_0/\delta \approx 0.057}$ (for ${\displaystyle \lambda =2}$).

Figure 3B shows the ranges of ${\displaystyle \beta /\delta }$ and ${\displaystyle {\beta }_0/\delta }$ for which the cooperative solution exists (as before, for ${\displaystyle \lambda =2}$). Solid lines of different colors correspond to three values of ${\displaystyle {\beta }_0}$ used in Figure 3A.

While the full set of our analytical results described in Appendix 1 is rather convoluted, here we present a simplified expression for the range of values of ${\displaystyle \beta /\delta }$, where the cooperative solution exists, and the corresponding range of ${\displaystyle m_{\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{p}}^{\ast }}$:

(12) $4\lambda <\frac{\beta }{\delta }<\frac{(\lambda -1)\delta }{4{\beta }_0}$

(13) $\frac{2\lambda \delta }{(\lambda -1){\beta }_0}<\frac{m_{\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{p}}^{\ast }}{m^{\ast }}<\frac{1}{2}$

These conditions were derived in the limit ${\displaystyle {\beta }_0/\delta \ll 1/\lambda }$ (see Appendix 1 for details).

Evolutionary dynamics

The competition for monomers is the main mechanism of natural selection operating in our system. The steady state with the lowest level of monomer concentration is favored by evolution since the competing states would not be able to proliferate at that level of monomers. Equation 13 implies that the monomer concentration ${\displaystyle m_{\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{p}}^{\ast }}$ in the cooperative solution is always less than a half of its value ${\displaystyle m^{\ast }}$ in the absence of the catalytic cleavage. Thus, once the cooperative state emerges, it drives out all chain sequences that still rely on non-catalytic cleavage for replication. The continuously increasing ‘fitness’ of the system can be quantified, for example, by the ratio ${\displaystyle m^{\ast }/m_{\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{p}}^{\ast }}$. Note that the non-cooperative solution has a fitness of 1, while the cooperative solution has a fitness higher than 2. Note that this fitness is defined at the level of the ecosystem comprising all the sequences in the chemostat and may not necessarily be attributable to individual members of that population. Over time, similar to microbial ecosystems, this population changes according to the laws of competitive exclusion (Gause, 1934; Tilman, 1982).

The fitness landscape of our system shown in Figure 4 depends on two parameters of the catalytic cleavage: β and λ. Its preeminent feature is a relatively narrow fitness ridge (orange and red color in Figure 4). For a fixed value of λ, this ridge corresponds to a sharp fitness maximum located at the lowest possible catalytic cleavage rate β for which the cooperative state still exists (see the red region in Figure 4B). In other words, for a given λ, the selective pressure would drive β down to the lower boundary separating cooperative and non-cooperative regions.

Figure 4

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However, λ and β are expected to co-evolve together. Indeed, ${\displaystyle \lambda >1}$ quantifies the ratio by which the structural properties of the type-${\displaystyle b}$ chain (its excess length, the hairpin unzipping free energy, etc.) slow down its replication compared to that of the type-${\displaystyle a}$ chain. Thus, it is reasonable to assume that λ could be easily modified in the course of the evolution. As a consequence of the selective pressure, both λ and β are expected to increase in the course of the evolution driving the system up the ridge in Figure 4. This intuition is confirmed by a direct numerical simulation in which we model the evolution as a Monte Carlo process with fitness playing the role of negative energy Parameters λ and β were allowed to vary randomly and independently of each other. A sample evolutionary trajectory is shown as a dashed line in Figure 4.

One can imagine several possible pathways leading to this self-sustaining cooperative system. (i) A pair of mutually complementary non-catalytic chains (${\displaystyle a}$ and ${\displaystyle \overline{a}}$ in our notation) gains function due to a copying error, giving rise to a small subpopulation of ‘sister’ chains (${\displaystyle b}$ and ${\displaystyle \overline{b}}$ in our notation) with nascent cleavage activity directed toward ${\displaystyle \overline{a}}$ and ${\displaystyle a}$, respectively. (ii) A pair of catalytically active chains ${\displaystyle b}$ and ${\displaystyle \overline{b}}$ emerges first, subsequently losing the catalytic inserts due to a copying error, thus giving rise to a small subpopulation of substrate chains ${\displaystyle a}$ and ${\displaystyle \overline{a}}$. According to the dynamic phase portrait of our system illustrated in Figure 2B, the second pathway is more plausible than the first one. Indeed, at least for the parameters used in Figure 2B, the conversion of only a few percent of ${\displaystyle b}$ chains to ${\displaystyle a}$ brings the systems into the basin of attraction of the cooperative fixed point marked green in the figure. The other scenario is less likely since it requires a much higher ratio of emergent over ancestral population sizes.

Our results indicate that the cooperative steady state emerges when the catalytic rate enhancement over the spontaneous cleavage rate is at least 10²–10³. This is a relatively modest gain compared to the 10⁹ enhancement reported for a highly optimized hammerhead ribozyme Scott et al., 2013. However, it is comparable to the rate enhancement observed after only five rounds of in vitro selection from an unbiased sample of random RNA sequences (Salehi-Ashtiani and Szostak, 2001). In the course of subsequent evolution, two main parameters of our model β and λ are expected to increase in tandem. The first parameter, β, is the catalytic cleavage rate, which, as we know, can increase over multiple orders of magnitude (Salehi-Ashtiani and Szostak, 2001). The second parameter, λ, quantifies the relative delay in the elongation of ${\displaystyle b}$-types chains compared to ${\displaystyle a}$-type chains. It may be caused at least in part by the difficulty of unzipping the hairpin, thus λ can be dramatically increased by making the hairpin stem longer. One should note that our model requires the catalytic activity of both ${\displaystyle b}$ chains and their complementary partner, ${\displaystyle \overline{b}}$ chains. Thus, sequences are expected to evolve to simultaneously optimize these two catalytic rates. The need for this compromise would likely prevent the catalysts from reaching the maximum efficiency observed, for example, in fully optimized ribozymes (Scott et al., 2013). The drive to further optimize cleavage activity might trigger a transition to more complex catalytic networks, for example, to an increase in the number of chains involved.

The proposed scenario is certainly not the only plausible pathway that could lead to the emergence of functional heteropolymers in the prebiotic world. To illustrate some of the unique features of the presented model, it is useful to place it in the context of other recent proposals. One of the most intriguing possibilities is the virtual circular genome (VCG) model recently proposed in Zhou et al., 2021. It is based on the observation that a relatively long ancestral genome can be stored as a collection of short overlapping RNA fragments of a circular master sequence or its complement. The model assumes unidirectional non-enzymatic replication of these fragments. On the one hand, it explains how a collection of relatively short RNA fragments (10–12 nucleotides each) could store a large amount of genetic information. On the other hand, computer simulations indicate that the VCG is susceptible to so-called sequence scrambling, where the appearance of repeats in the master sequence results in the loss of integrity of an entire circular genome (Chamanian and Higgs, 2022). The proliferation of the VCG is quite sensitive to sequence scrambling since the model assumes unidirectional polymerization, so that in order to return to copying a particular segment, one must copy the entire virtual circle.

Our model, while sharing some elements with the VCG model, differs in two important aspects: (i) it assumes that the non-enzymatic templated polymerization is bidirectional, and (ii) the functional activity of the heteropolymer is localized in a relatively short sequence region that is catalytically cleaved and thus replicated first. This implies that possible scrambling outside of the narrow functional region does not affect the viability of our autocatalytic system. Indeed, assuming the minimal hybridization length ${\displaystyle l_0=6}$ and random statistics of the master sequence, one gets the scrambling-free length ${\displaystyle \sqrt{2\cdot 4_0^l}+l_0\simeq 100}$. This is an order of magnitude larger than both ${\displaystyle l_0}$ and the length of the core region of the hammerhead ribozyme.

It may be possible to incorporate the selection mechanism proposed in this article into the VCG model. Such a hybrid approach would avoid the need for the biochemically problematic bidirectional growth, while explaining the emergence of early catalytic activity unaffected by sequence scrambling. This in turn could pave the way for the emergence of the rolling cycle, a model proposed as an alternative to VCG. (Tupper and Higgs, 2021; Rivera-Madrinan et al., 2022). The rolling cycle relies on strand displacement rather than cyclic melting of hybridized duplexes and requires a more sophisticated setup, including a pre-selected sequence pool and the availability of cleavage enzymes. Our model provides a plausible pathway for how this catalytic function could have evolved.

Our model aims to describe the early stages of the evolution of life on Earth based on non-enzymatic polymerization. While it may seem challenging to test it for conditions relevant to the origin of life, our main conclusions can still be verified experimentally. RNA or DNA can be used as model polymers in such experiments, as both have demonstrated catalytic abilities in cleavage reactions (de la Peña and García-Robles, 2010; Williams et al., 1995; Salehi-Ashtiani and Szostak, 2001; Breaker and Joyce, 1994; Silverman, 2005; Chandra et al., 2009). To simulate primordial polymerization driven by day/night cycling, the experiment would have to rely on enzymatic polymerization or ligation as used, for example, in PCR (Mullis and Faloona, 1987; Saiki et al., 1988) or ligase chain reaction (LCR) (Barany, 1991). However, it is important to note that our model assumes polymerization in both the 5′-to-3′ and 3′-to-5′ directions, unlike traditional PCR, which only adds new nucleotides in the 5′-to-3′ direction. This problem of bidirectional polymerization was solved by evolution using Okazaki fragments (Okazaki et al., 1968). Inspired by this discontinuous synthesis of the lagging strand of DNA, we propose a possible experimental implementation of our system based on ligation rather than polymerization enzymes. In this scenario, the system would be supplied with ultrashort random DNA segments. These segments, which are much shorter than the minimal primer length (${\displaystyle l_0}$), would play the role of ‘monomers’ and bidirectional primer extension would occur through a sequence of ligation steps connecting adjacent ultrashort segments to each other. Another important consideration for experimental implementation is the need to activate the nucleotides to provide free energy for polymerization. Thus, both the short fragments supplied to the system and the new primers formed by cleavage must be chemically activated.

The current manuscript is a theoretical study, so no data have been generated for this manuscript.

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Author details

1. Alexei V Tkachenko

   Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, United States

   Contribution

   Conceptualization, Software, Formal analysis, Investigation, Visualization, Writing – original draft, Writing – review and editing

   For correspondence

   oleksiyt@bnl.gov

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-1291-243X

2. Sergei Maslov

   1. Department of Bioengineering, University of Illinois Urbana-Champaign, Urbana, United States
   2. Department of Physics, University of Illinois Urbana-Champaign, Urbana, United States
   3. Carl R Woese Institute for Genomic Biology, University of Illinois Urbana-Champaign, Urbana, United States

   Contribution

   Conceptualization, Software, Formal analysis, Investigation, Visualization, Writing – original draft, Writing – review and editing

   For correspondence

   maslov@illinois.edu

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-3701-492X

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